Puzzle: worst luck holding AA

chapstick · #1 03-21-2005, 07:00 PM

I came across this old digest:

http://www.twoplustwo.com/digests/to...0_msg.html#284

and was intrigued by the discussion on when to fold AA preflop. Greg Raymer posted this

[ QUOTE ]

No fuzzy thinking. I wasn't commenting on the major question of the thread, just pointing out that the chances of AA winning (when all-in preflop) will never go below 20%. I guess the odds might go below 20% if there are enough players, but it takes more than 10, which is a number you'll never exceed in a casino game.

[/ QUOTE ]

which got me thinking about the following puzzle: What would be the worst possible table to be hold AA against assuming everyone goes all-in pre-flop?

My answer gives AA a chance of %3.91 to win. Does anyone have a lower result? Here's the table I devised: http://twodimes.net/h/?z=839505

pzhon · #2 03-21-2005, 08:29 PM

You can do worse if someone else has AA, too. Do you mean to exclude that case?

Against AA and lower pocket pairs blocking your flushes, AA can win 0% and tie 1.58%.

http://twodimes.net/h/?z=371717

A slight modification to break up the other aces gives 1.26% wins and 0.07% ties. I doubt that is optimal.

http://twodimes.net/h/?z=839649

chapstick · #3 03-21-2005, 08:42 PM

[ QUOTE ]
You can do worse if someone else has AA, too. Do you mean to exclude that case?

[/ QUOTE ]

Yes, sorry. I'm certain I had that written in, but I must have erased after I previewed.

chapstick · #4 03-21-2005, 08:51 PM

[ QUOTE ]

A slight modification to break up the other aces gives 1.26% wins and 0.07% ties. I doubt that is optimal.

http://twodimes.net/h/?z=839649

[/ QUOTE ]

Modifying yours, the lowest win rate I can find is 1.16%.

http://twodimes.net/h/?z=839688

But the tie rate is high.

pzhon · #5 03-21-2005, 11:25 PM

[ QUOTE ]

Modifying yours, the lowest win rate I can find is 1.16%.

http://twodimes.net/h/?z=839688

But the tie rate is high.

[/ QUOTE ]
Here is an example with 0.78% wins with 0.05% ties. I haven't tried much around it. The main modifications were to get rid of a couple of low pairs and add suited connectors.

http://twodimes.net/h/?z=839919

chapstick · #6 03-22-2005, 01:13 AM

[ QUOTE ]
Here is an example with 0.78% wins with 0.05% ties. I haven't tried much around it. The main modifications were to get rid of a couple of low pairs and add suited connectors.

http://twodimes.net/h/?z=839919

[/ QUOTE ]

I declare you the winner [img]/images/graemlins/wink.gif[/img]

This looks unsolvable on a computer. There are something like P(50,18)=10^29 possible deals to search through... can that be?

pzhon · #7 03-22-2005, 04:02 AM

Here is an improvement to 0.36% wins and 0.04% ties:

http://twodimes.net/h/?z=840376

The aces in the other suits are now suited, and I added suited connectors in the suits of the AA, and one offsuit connector.

This way, there is at least one card of each rank, so if there are trips on the board someone has quads, and if the board pairs, someone has at least trips. AA should not be able to win by a counterfeited 2-pair or overboat. How can AA win? There are still a few ways AA can win with one pair, and a few ways AA can win with a flush.

AA can tie with a straight, KQJTx on the board, where 2 of KQJT are hearts and 2 are diamonds, and the x is a spade or club. AA can also tie when there is a straight-flush on the board.

I think consideration of the possible methods of winning may be a way to prove that some arrangement like this is optimal.